So, if I have a civilization that lives in a hollow planet, with a sun in the middle. How would they tell time? The sun always shines, and it is a Constant Spring/summertime weather. They have animals that live about as long as they do on earth, trees, and other plants.

The planet will have a magical backstory. The magic also explains how everyone stays on the inside of the planet rather than falling into the star.

The history of this world is that it orbited a star, but for some reason (I'm not sure) it became a rogue planet. Most magic users of that time, who were considerably more powerful than the modern ones, banded together. They used all of their magic and sacrificed their lives to form a magical hollow planet. So now we can stop talking about the hollow earth theory, because it's not that relevant to the story. (I do acknowledge there are points in favor and against the hollow earth theory. I just read it online and found it interesting.)

There is no hole in the planet. It is possible through one set of caverns to go to the surface of the planet, but the only person who came back from there nearly suffocated/froze to death.

I just thought I would say that I don't think I'll pick a best answer or anything, because I plan on making several time systems, similar to how there is the metric and standard system of measurement. I will also probably use many different people's ideas.

$\begingroup$Keep in mind that there are areas on earth with very skewed day/night cycles, the north/south-pole and regions around it have 6-month day and 6-month night cycles. There are still animals there. Could be a good place to look for what would happen in a constant day situation.$\endgroup$
– SelenogJan 5 '16 at 16:02

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$\begingroup$As a less exotic alternative to the hollow world, a tide-locked planet would have much the same issues (although it would at least be possible to observe the stars on the night side, and possibly slight movement of the sun due to libration on the day side).$\endgroup$
– Ilmari KaronenJan 5 '16 at 19:32

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$\begingroup$Everything would fall into the sun, the the crust creates no net gravity inside the shell, (search on gravity inside a hollow shell) but if you want to suspend the laws of physics as we presently understand them, sure, go ahead. Spin would push everything to the equator, and also put massive strain on the crust. If you have time read the preface to Niven's Ringworld Engineers there's lots of good physics there.$\endgroup$
– JasenJan 6 '16 at 2:16

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$\begingroup$"Whether or not the earth is really hollow is a subject for debate" ... no.$\endgroup$
– YakkJan 6 '16 at 16:29

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$\begingroup$The day-light cycle is just one cycle on earth. Are there any other cycles on your hypothetical world? Seasons, tides, sleep/wake, menstrual, etc. Any notion of time would be based on whatever cycle seemed most immediate and important.$\endgroup$
– DCShannonJan 7 '16 at 2:51

20 Answers
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The primal civilizations evolving in such an environment would likely not develop any form of timekeeping. Without a natural daylight and season cycle there is little reason to.

As soon as technology and society develop in ways which makes timekeeping necessary for coordination of actions, some arbitrary fixed unit of time would likely be standardized. In the beginning it would be something natural like the heartbeat of a relaxed person or the time it takes for a rock to fall to the ground when dropped from eye-height. But the further society and technology advance, the more exact the definition will become. For convenience, let's call this unit "second" and assume that it is about as long as a second of our civilization.

Longer timespans would be measured in larger numbers of seconds.

The sleep-cycle of the average person would be around 100,000 seconds

Salaries and rents are paid every million seconds

A human would be considered an adult after 500 million seconds

A human lifetime would be about 2 billion seconds

When they use a decimal system, their life rhythms would likely be dominated by decimal intervals just because it's easier to handle. When their number-system uses a different base than 10, that base would likely dominate their life-rhythms.

Date-keeping would use an epoch-format counting the number of seconds since some important historical event and that count would be synchronized globally to allow precise dating of events. One would say "I was born at 56bil 234mil". A historian might say "the time between 22bil and 28bil was the age of chaos". A date for a meal would be like "Let's meet at 25-thousand" meaning that we meet the next time the last 5 digits of the calender are 25000.

When your civilization developed independently from humanity, it is of course very unlikely that their time-unit will be called "second" and even less likely that it will have exactly the length of a real-world second, but having it "quite similar to a real-world second" might make it more accessible to the reader.

$\begingroup$Great answer since that is exactly what we did with temperature. Degrees celcius start at 0 (when water freezes) and scales to 100 (when water boils). We could have picked anything else, but these we just convenient I guess.$\endgroup$
– Wim DeblauweJan 5 '16 at 19:52

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$\begingroup$Without a natural daylight and season cycle there is little reason to. - if we speak about homo sapiens sapiens, then there's still one cycle left - it is a menstrual cycle. Cannot find a reference now, but I read somewhere that some women were able to provide more precise time estimates (in comparison with men) during '70s-'80s several-months cave isolation experiments.$\endgroup$
– Vovanrock2002Jan 6 '16 at 8:30

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$\begingroup$I don't think it's reasonable that a group of people would evolve using "millions" or "billions" of anything as a reference to age/time. It's too many to count and too easy to lose track of where you are. We started with years because it was a reasonably long period of time to count, at a reasonably low number of them to count for a lifespan, while being short enough that we could split it into understandable shorter measures. The same has happened with foot/meter, cup/liter, temperature, etc. It's not typical to human nature count things in millions.$\endgroup$
– KeetaJan 6 '16 at 11:49

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$\begingroup$@Keeta There are just three countries in the world which do not use the metric system with milli- kilo- mega- etc. prefixes. Time is literally the only SI measurement which is not commonly expressed that way in most of the world, which is likely based on its origin of being bound to the day-rhythm and season-rhythm.$\endgroup$
– PhilippJan 6 '16 at 13:44

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$\begingroup$@Vovanrock2002 Now that's a strange thing. Because every woman I know has a different length of menstrual cycle, when not forcing it with hormones. It varies from the standard 28 by up to 4 days in normal women, and can vary by 21 days in unusual circumstances.$\endgroup$
– RedSonjaJan 7 '16 at 11:50

Telling time is a purely human concept. An animal sleeps when it is tired, hunts/eats when it is hungry, etc.

But humanity has come this far by working together, and that requires coordination, and thus a means of agreeing with others when a given event is supposed to happen.

(Never-mind that we are painfully aware of our own mortality and thus obsessed with the passing of time)

Earth's Conventions

Our current calendar / time arrangements are based on the Earth's movement in the solar system relative to the Sun. This obviously does not apply to your own world, however the need to "tell time" is still very much in demand.

Thus the people in your universe will reach an agreement, and a convention (or maybe multiple conventions based on their level of technology and communication between populations) will be reached.

A Gold Standard

If your world is relatively advanced, and people frequently travel between towns and cities, then you may have a gold standard which many people (such as all the people in a country, or on a continent) adhere to.

A magical clock could be built in the capital of your world by the best craftsmen/mages of that land, which keeps perfect time. Representatives from each town/city/country might travel there once a year to set their clocks which they would then take back home for everyone to use to set their local watches/clocks by (this could be a major event on your world).

This could be equivalent to our own New Year's, and the act of each representative safely transporting their clock to and from the capital could be an incredibly important undertaking. What is the clock is accidentally damaged, or moved such that the mechanism is thrown off? What if a rival town attacks their caravan and destroys their clock, or if a spy wanting to ruin their economy throws their main time-keeping piece off?

This could be a major mechanic of your world.

Smaller Scale Standards

Alternatively each region could have their own standard for when people sleep, etc., such as they did on ships in the old days: the man on watch would flip an hourglass. The punishment for failing to do so was very severe. All shifts on the ship would be measured by said hourglass.

In the end however, I can guarantee you that people would adapt and some arrangements would be made.

Implications of Eternal Day

A more relevant question might be what would the psychological and health implications of ever-lasting daylight be on people/animals. Which species would not exist on your world, and how would that impact other fauna? (such as the lack of a nocturnal predator to keep a population of other animals in check)

The solution here is easy: use a Tide Clock. If there are tides, you can use them to determine time. It's a daily cycle, and you can make life work around that.

Alternatively, you can simply define time by circadian rhythms that some important person undergoes. People will naturally feel sleepy, sleep, and wake up and thereby fall into a 24-25 hour pattern. Choose a king, and have his circadian rhythm define what time is. You have "Wakes" and "Sleeps", and not everyone needs to do this, but it will help.

$\begingroup$Having a moon in a hollow would would be difficult. I think the trick would be to get everyone's circadian rhythms to sync up so that you could have things like Business Hours. That's where standardized time would come in handy. Otherwise each individual would hit their own rhythm and gradually fall out of sync with everyone else.$\endgroup$
– AndyD273Jan 5 '16 at 15:48

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$\begingroup$@AndyD273 It doesn't seem too unreasonable to believe that a king's clock standard wouldn't get in the way of businesses. With everyone's agreement and mutual respect for that standard, if someone deviated then they might lose business. It could even be enforced and punishable.$\endgroup$
– The AnathemaJan 5 '16 at 15:59

$\begingroup$@TheAnathema That's what I was saying. You can't just define time by circadian rhythms, as they would be different from person to person. Having a "kings clock" would be the only real way to sync everyone up. In a daytime only world you'd need external zeitgebers to entrain everyone to the same cycle. (I like this question. I learned a bunch of new terms :)$\endgroup$
– AndyD273Jan 5 '16 at 16:05

$\begingroup$As I think about it, I feel that it is fair to say that this question could also apply on a tidally locked world, and you could get a lunar calendar in that situation.$\endgroup$
– AndyD273Jan 5 '16 at 16:06

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$\begingroup$@AndyD273 The moon does not have to be inside the hollow planet. If it would orbit the planet on the outside, it could still exert enough force on the bodies of water on the inside to create a tidal effect.$\endgroup$
– freekvdJan 6 '16 at 8:38

Humans are very good at recognizing patterns. When a pattern is noticed and able to be predicted, the patterns are often numbered or even named. This is how we currently have come to the measurement length of day, month, year, and even second.
Barring no sky, a human would look for the next most relevant predictable pattern in their life. It could be how fast a certain mushroom grows, how often a geyser erupts, or any other event.
There are huge variations in hollow earth theory. With each you can assume specific patterns to be seen.

You mentioned "hollow earth theory", though.

Assuming the model where the poles are open (Euler's theory), inhabitants would notice weather change (cold to warm) and wind direction change (north to south or south to north) each year. Their year would be the same as ours, with the same types of seasons as air exchanged from outer to inner earth. Although there may not be enough of an opening to fully see the stars, there would still be a concept of year, seasons, etc.

With Edmond Halley's hypothesis, there would be multiple concentric spheres. In this case, there would be a difference in spin speed between the spheres. This would be the most obvious pattern to choose for a standard for a "day" or "month".

Assuming the closed earth model, with a central "sun", the sun's position would wobble closer and further away from any one observation point (approximately) once a day due to the presence of our real moon. Tidal effects of the moon would still affect water that would be on the inside of the sphere. This pattern would generate the same month/day timescales we have now.

In all of these models, we have to assume there would be internal chunks of rock just floating around, "orbiting" the center at some rate. Depending on their period, they could be used as well.

All in all, if you are developing this for a story, I would say your best bet is to identify what patterns your "humans" would see based on the environment you create. Then, create time from that pattern.

$\begingroup$This article is the first time I heard of a hollow earth theory, but as an aside: At least the Halley version you depict does not exert gravitational forces to anything inside it (the "center of gravity" shell in the diagram is nonsense). The same annihilation of the gravitational forces prevents a naive Dyson Sphere from working. Cf the Shell Theorem page. Probably Euler's open version wouldn't work as expected either.$\endgroup$
– Peter A. SchneiderJan 14 '16 at 20:08

Our bottle

Long ago, Alice got tired of burning her lunch. At the same time, she noticed a kid fill a bottle with sand, then wait for the sand to run out. It took about the same time for the sand to run out of the bottle every time; perfect! Alice made a really big bottle, and put marks on the inside. One mark for "boil an egg", one mark for "grill a steak", and so on. Fill the bottle to the desired time, turn it over, and as soon as it ran out, the food was ready. Alice and her husband called it "our sand-bottle", which is where the term "hour" comes from: the time it took for the bottle to run empty at its highest setting.

Minute bottles

Then, Bob got the great idea to use water. Water trickled over a waterfall at a fairly constant rate, so he stuck a bottle under it. When the bottle was full, it would fall over; with a small machine to reset the bottle, it was a great timekeeper, since it didn't need to be reset. The bottle could be very minute; a "minute" bottle. It took 60 minute bottles falling over to make an our-bottle: 60 minutes in an our. Even better, a stack of numbered bottles meant the first real clock had been invented.

Clonk towers

For a long time, "ours" and "minutes" were the de facto standard. That is, until a wizard noticed that when water flowed faster, his minute bottles fell faster. He got tired of time measurements varying when he was at his lab and when he was at home, so he built a tall tower with a magical light at the top. Every time a bottle fell, a light was added, 60 lights in total. Perfect! He could easily measure minutes and ours by looking out a window. He decided these would be named "clonks", due to the noise they made when changing lights.

Days and Time Zones

Soon, the next town wanted one. So, he built another light tower, only this time, instead of hooking it up to a bottle, he set up a sensor that watched the other tower blink. That way, both towers were synchronized. There was a little bit of lag, but no one really noticed. Eventually, he realized that everyone would want a tower, so he set off on a great journey, to place clonk towers in a ring around his world. However, he realized that the lag would be an issue; to fix the problem, he decided to to set up a series of small clonk towers, then an ominously huge clonk tower, or O-clonk. Each huge tower would be one "our" apart. His friends begged him to use an even number, but he decided that 24 "ours" around the planet would be to his liking. His clonk towers were a huge success, and soon people everywhere knew if it was 5 O-Clonk. The time it took for a signal to go around the world was known as a "day", or so the wizard said. No one knows why he called it that. Wizards are weird that way.

Second

Finally, as time went on, people needed finer measurements. Someone decided to add a second set of even smaller bottles to the first; finally, the "second" had been invented. Some wise-guy made a series of 1000 bottles that took only 1 second to tip over, but his "many-second" never really caught on.

Other units of time

Other units of time sprang up naturally: going 7 "days" without any food made one "weak", it took roughly 30 "days" to recover from "mumps", a famous comedian always some 365 "days" to come up with a new "jeer", and so on.

$\begingroup$Wow, that is the puniest thing I have ever read in my life. I don't think time zones would be realistic, because people would want to know when things are happening in other places. Time zones were developed because the sun came up at a different time when you went east or west from a location. Although, The hourglass type idea is pretty good.$\endgroup$
– Xandar The ZenonJan 6 '16 at 21:52

$\begingroup$Well, at least in this scenario, the wizard invented time zones out of necessity - his towers lagged. He (arbitrarily) decided to use the 24 "our" day, which means each slice of the world uses its own O-Clonk, resulting in 24 'time zones'. He could have instead added an "our" to each O-clonk to keep them in perfect sync, of course.$\endgroup$
– ArmanXJan 6 '16 at 22:10

$\begingroup$I'm sure to laugh, but it might take a few "ours"$\endgroup$
– Redwolf ProgramsApr 18 '18 at 2:50

I would say things would start by using natural rhythms. So to start having some concept of time you need to observe changes in SOMETHING. On our planet the easiest is the sun rising and setting and the seasons coming and going.

On this planet it would likely be biological rhythms. The life of a flower, plant, tree etc. How long a child takes to get his/her first tooth etc. Part of the need to track time is to know when things are going to happen and plan for it. Knowing when the rainy season is, or winter so you can store foods.

So WHAT do they need to 'time' will have a huge impact on how they tell time. Does certain fruit ripen on a schedule? what about vegetables? Are there rhythms to animal breeding? All these can be used.

As far as a day/night cycle type timers it might be harder unless for some reason animals have an active/inactive cycle for rest, not necessarily all animals would have the same cycle, but say a herd of Elk might have a consistent, active inactive cycle that is fairly consistent, which could be used as a rough clock.

$\begingroup$All of the rhythms you talk about here are only biological rhythms built into the underlying biological machinery of life because of the solar cycle. Remove the solar cycle from ever having been part of the equation and this answer collapses. Why do animals breed when they do? Well, because winter is a bad time, so do it as early as possible so that when winter comes again, the young are old enough to survive. Except "winter" is a concept that only exists because of the solar cycle. Same with tides: there's no moon. And so on down the chain.$\endgroup$
– Draco18sJan 5 '16 at 16:11

$\begingroup$@Draco18s I know. however, that doesn't mean they won't create their own rhythms, and for a story this can more easily just be 'how it is'. Since it is impossible to 'live' on the inside of a sphere like this anyway. Everything would fall into the light...$\endgroup$
– bowlturnerJan 5 '16 at 16:21

$\begingroup$The circadian rhythm exists as a biological "backup signal" for when we can't see the sun. It's a timer baked into our biomachinery due to the presence of the sun. It's less accurate, but good enough for cloudy days or deep cave living. So if the sun was always out, such a rhythm would not develop. Or rather, it wouldn't be based on the period of sunrise to sunrise. It would be based on some other natural cycle. The asker is trying to to determine what that other cycle is that would cause the biomachinery to develop in the first place, necessitating society's need for a more accurate clock.$\endgroup$
– Draco18sJan 5 '16 at 16:24

$\begingroup$@Draco18s I see realitycheck and society tags. Not hard science. It is much easier to just assume that plants and animals have come up with some kind of rhythm that can be counted on. You only care about the passage of time if you care about some upcoming semi predicable event.$\endgroup$
– bowlturnerJan 5 '16 at 16:30

$\begingroup$@bowlturner actually, there would be no gravity in the hollow sphere, but centrifugal force of the planet spinning would keep you glued to the ground.$\endgroup$
– Xandar The ZenonJan 5 '16 at 18:58

First we need to define why time is useful. Time useful to people for a few fundamental reasons.

Planning

On earth it is useful to know when winter is coming/ending so I can plan how much food to store and when I can plant new crops.

People want to gather at specific times/days/weeks for events/festivals/competitions/etc.

I know the train will arrive at noon so I need to leave my house at 11:30 to reach the train station.

Coordination

Let's meet at this rock at this time.

I'll create a diversion on this side of town so you can do XYZ on the other side of town.

Let everyone know that the train will arrive at noon so we can board the maximum amount of passengers in a short window.

Navigation

Accurate time keeping was the key that unlocked accurate longitudinal position fixes.

Record keeping

The king was crowned 50 years ago at the age of 18.

The extent that you integrate these elements in your story will drive what kind of time keeping is needed for your characters.

Maybe the constant sunlight creates regular rain/weather cycles, and these daily rain showers would be useful to count. That would most likely be good for the more macroscopic events, ie measuring a person's age, scheduling festivals/events/etc. Of course the rain/weather cycles would be driven by terrain too, but you haven't given us many details on the terrain in your world.

The more fine grained events like a train schedule and/or navigation can be maintained via spring wound or pendulum driven town clocks. These are tried and true mechanisms, depending on the tech level for your world, that work well on land. Any oceans that are present on your world would probably be devoid of tides/swells so they might work well on the ocean too.

All that being said you can avoid time entirely by being clever if you want.

Need to create a diversion after everyone is in place?

Everyone can gather at one spot then travel as a group dropping people at their location until the final actor is in place to create the diversion. The diversion is big enough to be noticed by all actors so the triggering of the diversion sparks their independent actions. Plot tension ensues when one of the actors thinks everyone should've been in place already...

Have a runner(s) coordinate message passing that everyone is in place. Plot tension when an actor questions if the runner was captured on the way to deliver the message...

Need to navigate to spot X?

Go north/south to the correct latitude and follow that line east/west to your destination. Plot tension ensues because people know this technique and might be waiting in ambush on the latitude line.

Use a local guide and/or guides for navigation. Plot tension centers around the trustworthiness of the guide.

Terrain isn't uniform and maps with landmarks are available. Plot tension around the accuracy of the map.

Festivals are launched at the whim of the king/queen/important person and a town crier is used to spread the word.

Record keeping is only done by sequential entry into log books. Era's marked by lifespan of figurehead, or history is squelched so unpleasant comparisons aren't made against the current figurehead.

$\begingroup$"First we need to define why time is useful." Time is useful because it keeps everything from happening all at once. :P Perhaps you meant 'why tracking time is useful'...$\endgroup$
– Xavon_WrentaileApr 10 '16 at 7:30

In order to worry about timekeeping, I am assuming you have some sort of civilization that is at least somewhat advanced enough to worry about time.

Because time could have begun to be observed most likely when people are tired and when people become hungry and from there timing of events become popular.

As for timekeeping, there are ancient ways of timekeeping that did very well in ancient times: the basic hourglass of sand, or the interesting variations of water clocks. And from there you could extrapolate further inventions of timekeeping in accordance to how advanced your world's current technology is.

Water

If you have major river, there aren't seasonal changes or snowmelt, so it is from a flowing artesian spring. This means it is a nearly constant flow.

Your early civilizations start noticing that it takes this many turns of a water-wheel (or previously, a stick to get from A to B). "Wow Blorka has been gone for 92 turns of the water wheel. Hope she's okay..."

Not too long after that, it was standardized to avoid the imperfections, much like we standardized distances and size. Now, "OMG Roberta has been gone for like almost a hundred turns. Text that girl and ask what's taking so long."

People would keep time by how long it takes to do something or how long it takes for something to happen. For example, in old medicinal cures, there was sometimes a step "sing three Ave Marias" which modern doctors might dismiss as being "obviously" not an essential part of the treatment. However, what might be important for the treatment is the time elapsed while singing three Ave Marias, which could be easily measured according to something everyone knew the duration of, in an age when accurate watches and clocks were not handy. (There's a bit more about this in Radiolab's "Staph Retreat.")

Gravity-driven clocks like sand or water or a pendulum bob falling, or chemical processes (candles burning is just one suggestion in some of the other answers), or anything else that takes a consistent amount of time to happen, could also be used. Our modern second is based on oscillations of a Cesium atom that are very consistent under specified conditions.

If there are other periodic phenomenon important in their environment, the principal time units would likely be the periods of that phenomena. Here on Earth, our major units are the period of rotation around the planetary axis and orbit around the star because different parts of those cycles have significant consequences that affect survival odds and beneficial conditions for other activities, providing evolutionary advantage to being able to predict those periodic changes.

I'm reminded of the "sun squares" from Ringworld (Niven). Ringworld's enormous sun squares are connected in a circle by a super-strong thread of some sort and the whole chain rotates slowly around the sun, casting shade and giving the Ringworld the appearance of regular night and day. Assuming your sun is tiny and suspended in the centre of the hollow sphere, could you adapt this idea? Then your characters would have a basis for inventing time. (Bonus: you have another target for your antagonists to attack.)

Nightfall (Asimov/Silverberg) is a good example of a world with continual day, though it's due to the planet having six suns. The inhabitants appear to be quite human-like. They sleep and have a concept of breakfast time and so on, but 'darkness' is a theoretical concept only obtained under artificial conditions. I seem to recall it's not specified in the story exactly how time was invented under perma-light conditions, and in any case it's not important except for predicting the date of impending doom.

$\begingroup$Why would a world with no nighttime even have candles?$\endgroup$
– mg30rgJan 5 '16 at 15:37

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$\begingroup$@mg30rg Indeed that is what they'll do during the period of development that mirrors our 'caveman' era but eventually as they progress technologically they'll need larger indoor structures and something to do with all the rendered fat from processing animals for food/clothing. It's likely they'll burn it to cook with but some bright spark might think to take a small section from the fire to light his way to bed...$\endgroup$
– Scott DowneyJan 5 '16 at 16:07

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$\begingroup$@mg30rg What about cave exploration? Or basements? Then they'd need candles.$\endgroup$
– Xandar The ZenonJan 5 '16 at 19:01

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$\begingroup$@XandarTheZenon I would still vote for mirrors. In case of permanent daylight they are the ultimate solution to light shady places.$\endgroup$
– mg30rgJan 6 '16 at 8:27

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$\begingroup$@mg30rg While you are correct in the case of basements and other such areas, you would still need it for cave exploration. While magic light orbs would provide most of your light, if you had a candle and it stopped burning, you would know when the air was dangerous.$\endgroup$
– Xandar The ZenonJan 6 '16 at 20:37

You would measure time by the number of bathroom runs; number ones and number twos. A standardized number two period equals about 10 number one periods which is handy, in a manner od speaking. Engineers wanted to do a time reform but couldn't even agree among themselves whether 8 or 16 would be a better fit.

People would say "He had a very long night. He had already taken 3 leaks and was lying awake thinking about a fourth one."

Or "Boy that was a long night out, I think it was at least 6 leaks." Commonplace experience is that the time needed to drink one pint of beer fairly exactly amounts to one leak. This identity is astonishingly independent of the speed at which you drink the beer, a counterintuitive fact known as time dilation at higher drinking speeds.

The number of leaks seems to increase with increasing masses of people (->Oktoberfest) and strange attractors. As is detailed below, both are also connected to memory leaks.

Most other aspects of reality tend to become distorted in the vicinity of leaks induced by alcohol. Examples are space which appears warped after more than 4 leaks, apparent increases in radiating beauty which may actually be a secondary effect of pupil dilation, etc.

Scientists are still debating whether money leaks are a primary effect occuring in these circumstances or a secondary effect of ordinary and memory leaks. Memory leaks have been unequivocally shown to be entangled on the quantum level with ordinary leaks. They are harder to measure than money leaks though, because in a law-of-conservation-defying way the leaking memory doesn't seem to go anywhere. (Much of it gets communicated, no doubt, but underlies the same leaking on the recipient side.) Because memory leaks stay ipso facto often unrecognized their prevalence is unclear. While a lot of active research is going on in the field it has so far failed to produce meaningful results, if any.

$\begingroup$I'm sorry, but that's kind of disgusting, and I don't think any civilized person would measure time by their bathroom breaks.$\endgroup$
– Xandar The ZenonJan 6 '16 at 20:40

$\begingroup$I think Americans are a bit more squeamish than Europeans about body functions, but under the obvious humour lingers a serious idea: Body functions in general are an obvious time scale, from eating and sleeping over menstruating and carrying babies to aging. Not exact, but very obvious and natural. It's noon not at 12 but when it's time to eat the mid wake time meal. Small communities (a big farm) would eat and possibly sleep together or in sub groups, partly for practical and partly for physiological reasons. Thus these things would structure their time.$\endgroup$
– Peter A. SchneiderJan 6 '16 at 21:08

$\begingroup$Oh, and I forgot one of the most interesting body function for measuring time, the heart beat. Universally structuring time by body functions may have more profound implications on culture:Considering "Know ye not that ye are the temple of God" (are you not from Utah?) one could easily imagine a religion which considers the web of time as spun by God through all living beings etc. This would lead to a much more organic (for example, more flexible, less universal) concept of time compared to celestial or mechanical ones.$\endgroup$
– Peter A. SchneiderJan 7 '16 at 8:49

$\begingroup$Ctd. A religion under these circumstances (no celestial bodies, "distributed" time paradigm) could also be more "distributed" and never produce a universal creator. Also consider Kant's famous analogy "the starry heavens above me and the moral law within me": Even moral laws may be more "distributed" (less universal, less abstract) than on earth due to a lack of "universal" (so to speak) absolutes in the perceived reality$\endgroup$
– Peter A. SchneiderJan 7 '16 at 9:01

Modern clocks don't use solar cycles to measure time. Atomic clock uses Cesium radiation to define how "long" a second is. So solar cycles isn't a must if you are advanced enough and that's where magic comes in (you've stated that your antagonists are capable of it). Depending on the level of magic use (capability to observe certain periodical constants) it is entirely possible to measure time without day and night cycles.

$\begingroup$Possible to measure time, sure, but what units of measurement would have meaning? A second is defined as 9,192,631,770 (nine billion) oscillations of Cesium. That's hardly a useful measure of time prior to being able to detect a frequency that high. We need something slower and visible to the naked eye.$\endgroup$
– Draco18sJan 5 '16 at 16:14

I interpret your question so that there are no "celestial" cycles whatsoever (so also e.g. no moon phases, no tides, etc.)

The first thing one notices is that no external cycles does not mean no cycles at all. Consider your heartbeat: That's a cycle not because of something external, but simply because pumping blood using contraction of muscles inevitably leads to cycles. Also, in our brain waves we see certain frequencies, which are also likely because the brain works better that way — after all, also our computers are based on a clock, not because computation is somehow related to clocks, but simply because it is much easier to organize the complex processes if you have a common clock to synchronize on.

However, what about shared clocks? Well, the one development that gave evolution a major boost was sexual reproduction. And especially simple organisms tend to synchronize their fertility phases so that the chance to reproduce is increased. Now in our world this is bound to the astronomical cycles (mostly the seasons). But in a world without "external" cycles, I'd expect this to be regulated by things like pheromones (after all, the equivalent mechanism, hormones, synchronizes the different circadian cycles in our body, too). Since e.g. the reproductive cycle of plants will affect the availability of food (pollen, fruits), animals will synchronize their cycles to them.

In short, I'd expect that world still to have cycles, however it will probably be a more complex network of interlocked cycles, and many of the cycles will probably be of a more local nature (e.g. in our world the seasons are essentially the same on the whole hemishpere, although the effects differ depending on latitude; in your world different parts may have different "seasons"; even different length of seasons).

Therefore people in your world will be used to cycles just as we are, just that the cycles are of biological rather than astronomical nature.

Note also that most early clocks other than sundials didn't use cyclic mechanisms, but processes that go one way and have to be manually restarted. A candle clock measures time by how much of the candle burned away. A sand clock measures time by how much of the sand has flown to the lower container. Same about a water clock. All of those would be experienced by the people in your world just the same as in our world.

So in short, people would originally tell time based on biological/ecological cycles, and then move on to clocks just as in our world.

Any regularly reoccuring event could convey the concept of time. I recently read a story where the main character was able to break out of a pocket universe created by a wizard who made the mistake of not accounting for time. Anyone who entered was stuck because they would quickly loose their concept of time. The main character solved the problem by creating a water clock (it was raining in this pocket universe) using a cup and a stick on a pivot point. It would balance when the cup was empty but, as it filled with rain water, it would tip over, emptying the cup and thus resetting the balance. Since it did this over and over again with acceptable regularity, it kickstarted time in this pocket universe.

Consider this: You have a gas giant. In orbit around the gas giant is a molten moon, much like how earth's moon was molten until it had cooled sufficiently.

The Hollow earth shell rotates around this molten core at a rate of one rotation per day.

Just like our moon, the heavier elements have gravitated to the side of the moon that is closest to the gas giant (tidal lock). The heavier and colder elements on the giant side of the core produce about as much light as the Earth's moon reflects during our night time.

This gives you a night cycle when you are closer to the gas giant and a day cycle when you are farther away.

There are more "heavy" elements than "lighter". This means that the cold region wraps around the warmer region like an uppercase "C". This means that the "north and south" poles of the shell are in a constant "night". Freezing temperatures are common near the poles.

Also near the poles you would expect the gravity to be weaker, atmospheric pressure would be reduced, and breathing would become difficult. Living creatures would experience sickness from the Coriolis effect. Not someplace people would want to visit often or at all.

As mentioned in other answers, this would have to be an artificially created world for the shell to not be torn apart from the centrifugal forces. More than likely the creators also put in place a way for the inhabitants to refuel the cooling core at the center. This gives more reasonable motivations for your antagonist to say "All things must come to and end" instead of just ending the world "For the Evulz"

All the protagonist must do is reach the refuel station and ensure that it saves the dying core before the "tipping point of no return" the refuel station is likely near a pole on the shell since it has easier access to the core due to the reduced centrifugal forces. Also, the station likely receives fuel from the gas giant.

I also didn't mention the gravitational effects from the gas giant at night, but I'm a little fuzzy on what those would be exactly.

You don't need time until there is something periodic that you have to measure against. With no day/night cycle and no seasonal cycle, there are only two things I can think of that a human needs cyclicly (at least in a primitive (tribal) society)

Sleep cycles. Humans even in primitive societies sleep about 6 to 7 hours a day (often with an hour of wakefullness in the middle). So let's say roughly 1/3 of their time is for sleeping. Such a society is likely to come up with a "day" measured in "sleep"s. Each "day" is about 3 sleeps long. Not very precise, but it would serve their needs.

Menstrual cycles. Human females start menstruating roughly every 28 days, and are typically done with it after the 7th day. Women who live together closely (like in a family, band or small tribe) tend to match cycles too. For obvious reasons, males often find this a good time to go hunting. So its worth keeping track of. Due to the math here, it looks fairly natural to make "weeks" based on the 7-day period, and "months" based the roughly 4-week menstrual cycle.

(One might further speculate that this is the true source of 7-day weeks and 27-31 day months, not the moon, but that's beyond the scope of this stack, and likely unproveable).

Once society develops more complex behaviors that are time-based (eg: paid labor, longitudinal navagation, whatever), it might need finer and more accurate measures of time. But you'd expect that when that happens they'd build on their old imperfect systems (like humans here on earth did).

Pellucidar, the quintessential hollow earth of adventure fiction created by Edgar Rice Burroughs, operates essentially as you describe, with a perpetual noon-day sun at its center. The denizens of Pellucidar, which include sentient, psychic Rhamphorhynchids(Mahars), ape-men (Sagoths), and anatomically modern humans (Gilaks) do not seem to know or care about the passage of time at all. When two earth men drill down into the inner world, they dwell there for years without aging.

However, the main character does impose a system of time on Pellucidar based on the rotation of a small moon that floats fixed between the inner sun and the surface of the inner world. From the novel "Pellucidar" (1915) comes the following description:

As I watched it, I saw that it was revolving upon an axis that lay
parallel to the surface of Pellucidar, so that during each revolution
its entire surface was once exposed to the world below and once bathed
in the heat of the great sun above...

Here I saw a chance to give time to Pellucidar, using this mighty
clock, revolving perpetually in the heavens, to record the passage of
the hours for the earth below. Here should be located an observatory,
from which might be flashed by wireless to every corner of the empire
the correct time once each day. That this time would be easily
measured I had no doubt, since so plain were the landmarks upon the
under surface of the satellite that it would be but necessary to erect
a simple instrument and mark the instant of passage of a given
landmark across the instrument.

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